Anomalous diffusion, fractal reaction kinetics, and Alzheimer’s disease

Anomalous diffusion of molecules in heterogeneous media (like a membrane of a living cell) is known to alter classical mass-action rule of reaction kinetics to so-called fractal reactions, where sub-linear dependence on time of mean square displacement of a random walker induces “memories” in the reaction rate (Kopelman 1988). Here we theoretically examine alternative (competitive) processing of a substrate molecule by two enzyme species in a heterogeneous media. We found that difference in the degrees of diffusive abnormality of enzyme species largely affects which pathway dominates in equilibrium outflow of the alternative processing. To derives these results we applied 1) the scaling theory, 2) age-structured kinetics, and 3) Monte Carlo simulation, among which 2) is new in the literature to our knowledge. Possible application of our theory to the onset of the accumulation of beta-amyloids, that is highly associated with the Alzheimer’s disease, is discussed. Kopelman, R. (1988) Fractal reaction kinetics. Science 241, 1620-1626